chap2-3slidespt3

chap2-3slidespt3 - Axiomatic approach Summary of theorems...

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Axiomatic approach Summary of theorems For any A , B ∈ F : 1 P ( A ) = 1 - P ( A c ) (very useful in combination with DeMorgan’s) 2 If A B then P ( B - A ) = P ( B ) - P ( A ) 3 P ( A ) = P ( AB ) + P ( AB c ) 4 P ( A B ) = P ( A ) + P ( B ) - P ( AB ) ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 82
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Discrete Probability Spaces Definition A probability space ( S, F ,P ) is discrete if: S is finite, or S is countably infinite A discrete space is either (i) Finite; (ii) Countably infinite. ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 83
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Finite Probability Spaces Sample space The sample space S is of the form: S = { s 1 ,s 2 ,...,s N } where N is a positive integer; s i is the i th possible outcome. ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 84
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Finite Probability Spaces Events algebra It is usually not advantageous nor necessary to exclude certain subsets of S from F . Therefore, F = P S ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 85
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Finite Probability Spaces Probability function Standard definition of P ( · ) : Through probability masses p i , i = 1 ,...,N . To each s i S , i = 1 ,...,N , we associate a real number p i : 1 p i 0 , i = 1 ,...,N 2 N i =1 p i = 1 ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 86
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Finite Probability Spaces Probability function For any event A ∈ F we define P ( A ) = X s i A p i Example: A = { s 1 ,s 4 ,s 6 } ⇒ P ( A ) = p 1 + p 4 + p 6 . Note: p i = P ( { s i } ) ,i = 1 ,...,N . ECSE 305 - Winter 2012 (slides based in part on the notes by B. Champagne) 87
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Finite Probability Spaces Probability function P ( · ) satisfies the probability Axioms: 1 Since p i 0 , i = 1 ,...,N : P ( A ) = X s i A p i 0 2 Since N
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chap2-3slidespt3 - Axiomatic approach Summary of theorems...

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